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JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Matrices

Let A be a matrix of order

3 \times 3

and

∣ A ∣ = 5

. If

∣2 adj (3 A adj (2 A)) ∣ = 2^{α} \cdot 3^{β} \cdot 5^{γ}

where

α, β, γ \in N

, then

α + β + γ

is equal to

(A)25
(B)26
(C)27✓ Correct
(D)28

Answer: (C)

Solution

For a

3 \times 3

matrix,

∣ k A ∣ = k^{3} ∣ A ∣

and

∣ adj A ∣ = ∣ A ∣^{2}

. Working outward,

∣2 adj (3 A adj (2 A)) ∣ = 2^{3} ∣3 A adj (2 A) ∣^{2} = 2^{3} (3^{3})^{2} ∣ A ∣^{2} ∣ adj (2 A) ∣^{2} .

Since

∣ adj (2 A) ∣ = ∣2 A ∣^{2} = (2^{3} ∣ A ∣)^{2}

, this equals

2^{3} \cdot 3^{6} ∣ A ∣^{2} (2^{6} ∣ A ∣^{2})^{2} = 2^{15} 3^{6} ∣ A ∣^{6}

. With

∣ A ∣ = 5

it becomes

2^{15} 3^{6} 5^{6}

, so

α = 15, β = 6, γ = 6

and

α + β + γ = 27

.

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Single correct option

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